Here, we derive closed-form expressions for the minimum coupling strength and link density necessary to have a stable synchronisation – as well as an upper limit to the chaoticity that can be synchronisable – of diffusively-coupled, identical maps, in generic regular graphs and class-specific cyclic graphs. Cyclic graphs are lattices having cyclical node-permutation symmetry (implying periodic boundary conditions and identical node neighbourhoods). In particular, we make derivations for \(k\) -cycles (also known as Wiley-Strogatz-Girvan networks [1]}) and \(k\) -Möbius ladders (non-planar graphs), which we introduce in this work to extend the classic Möbius ladder with degree 3 [2]}, [3]} to higher degrees. Our finite-size results show striking differences between these 2 cyclic graphs, only becoming similar when converging to the complete (all-to-all) graph. Moreover, we show that our expressions can change for different degrees and in the thermodynamic limit (i.e., infinite system size). Our derivations are based on the MSF [4]} and the graph's Laplacian eigenvalues (focusing on the smallest non-zero and largest eigenvalue), making our approach general. Overall, our work complements the general understanding of synchronisation phenomena in CMLs and provides detailed mathematical derivations leading to exact analytical results.
